Serre's uniformity problem for mod-p Galois representations of elliptic curves over Q and uniform finiteness of rational points on Y_0(N) over number fields: the open core of the four questions of Bourbaki expose 511

Serre's Bourbaki expose 511 (1977/78), reporting Mazur's theorems on rational points of the modular curves $X_0(N)$, ends with four questions. Two were subsequently resolved by Kenku: the optimal constant $C=8$ for the size of a $\mathbb{Q}$-isogeny class of elliptic curves (1982, and sharp), and the determination of the rational points on $Y_0(M)$ for composite levels $M$ (1979-1982, completing the list of rational isogeny degrees over $\mathbb{Q}$). The other two remain open and form the open core. The first is Serre's uniformity problem: is the mod-$p$ Galois representation $\rho_{E,p}:G_{\mathbb{Q}}\to\mathrm{GL}_2(\mathbb{F}_p)$ of every elliptic curve $E/\mathbb{Q}$ without complex multiplication surjective for every prime $p\geq 41$? The published reductions show that, for $p>37$, a non-surjective image must be contained in the normalizer of a non-split Cartan subgroup, so the problem is now exactly to show that the modular curves $X_{ns}^+(p)$ have no non-CM rational points for large $p$. The second asks whether, for every number field $K$, there is a finite set $S_K$ of primes such that $Y_0(N)(K)$ contains only complex multiplication points for prime $N\notin S_K$; this is known only under the generalized Riemann hypothesis, and unconditionally only for $K=\mathbb{Q}$. The objective is to determine whether Serre's uniformity problem (expose Question 2) and the uniform-finiteness assertion over number fields (expose Question 4) hold.

Background

An elliptic curve over $\mathbb{Q}$ is a smooth projective curve of genus 1 together with a rational point (the origin); its rational points form an abelian group. An isogeny between elliptic curves is a nonzero homomorphism of group varieties, equivalently a finite surjective morphism; it is cyclic of degree $N$ if its kernel is cyclic of order $N$. Two elliptic curves over $\mathbb{Q}$ lie in the same $\mathbb{Q}$-isogeny class if they are joined by an isogeny defined over $\mathbb{Q}$. The modular curve $Y_0(N)$ is the coarse moduli space of pairs $(E,A)$ with $E$ an elliptic curve and $A$ a cyclic subgroup of order $N$: a $\mathbb{Q}$-rational point of $Y_0(N)$ is exactly a pair $(E,A)$ with $E$ defined over $\mathbb{Q}$ and $A$ rational over $\mathbb{Q}$, i.e. a cyclic $N$-isogeny defined over $\mathbb{Q}$ (up to the choice of quotient or subobject). The smooth compactification $X_0(N)$ adds finitely many points at infinity called cusps. An elliptic curve has complex multiplication (CM) if its geometric endomorphism ring $\mathrm{End}_{\overline{\mathbb{Q}}}(E)$ is an order in an imaginary quadratic field rather than $\mathbb{Z}$; the corresponding points of a modular curve are called CM points, and all other points (equivalently, all pairs with $\mathrm{End}_{\overline{\mathbb{Q}}}(E)=\mathbb{Z}$) are called non-CM.

Serre's expose 511 reported, with proofs and refinements, Mazur's theorems: (Theoreme 1) for $N$ prime outside $\{2,3,5,7,11,13,17,19,37,43,67,163\}$ the set $Y_0(N)(\mathbb{Q})$ is empty, so those twelve primes are exactly the prime degrees of rational cyclic isogenies; (Theoreme 2) there exists a constant $C$ such that each $\mathbb{Q}$-isogeny class of elliptic curves contains at most $C$ $\mathbb{Q}$-isomorphism classes; (Theoreme 5) if $E/\mathbb{Q}$ is semistable (it has good or multiplicative reduction at every prime) and $N\geq 11$ is prime, then the Galois representation on the $N$-torsion of $E$ is surjective. For the last statement: for a prime $p$, the $p$-torsion $E[p]$ is a two-dimensional $\mathbb{F}_p$-vector space, and the natural action of the absolute Galois group $G_{\mathbb{Q}}=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on it is the mod-$p$ Galois representation $\rho_{E,p}:G_{\mathbb{Q}}\to\mathrm{GL}_2(\mathbb{F}_p)$. Serre had proved in 1972 that for non-CM $E/\mathbb{Q}$ this representation is surjective for all but finitely many $p$, the exceptional set depending on $E$; uniformity asks for a bound independent of $E$.

The maximal tori of $\mathrm{GL}_2(\mathbb{F}_p)$ are its Cartan subgroups: the split Cartan (diagonal matrices) and the non-split Cartan (the image of $\mathbb{F}_{p^2}^{\times}$, irreducible as a two-dimensional representation); adjoining the involution that swaps, respectively conjugates, the eigenlines gives their normalizers $N_s(p)$ and $N_{ns}(p)$. To each proper subgroup $G$ of $\mathrm{GL}_2(\mathbb{F}_p)$ satisfying mild conditions (surjective determinant, containing $-I$, and containing an element with trace 0 and determinant $-1$) one attaches a modular curve $X_G$ with a finite morphism $\pi_G$ to the $j$-line, the affine line whose coordinate is the $j$-invariant of an elliptic curve (the modulus classifying elliptic curves up to $\overline{\mathbb{Q}}$-isomorphism), such that, for every $E/\mathbb{Q}$ with $j(E)\notin\{0,1728\}$ (both excluded values are CM, so the exclusion is harmless for the questions below), $\rho_{E,p}$ is conjugate into $G$ if and only if $j(E)=\pi_G(P)$ for some $P\in X_G(\mathbb{Q})$. In particular $X_{ns}^+(p)$ denotes the curve attached to the normalizer of a non-split Cartan, and $X_s^+(p)$ that attached to the normalizer of a split Cartan; the split-Cartan normalizer is conjugate to the group defining the Atkin-Lehner (Fricke) quotient $X_0^+(p^2)$ of $X_0(p^2)$, so $X_s^+(p)\simeq X_0^+(p^2)$.

Over a general number field $K$ three further inputs appear. The generalized Riemann hypothesis (GRH) asserts that the non-trivial zeros of every Hecke $L$-function over $K$ (the analogues of the Riemann zeta function) lie on the critical line; several strongest-known results below are conditional on it. The Hilbert class field of an imaginary quadratic field $F$ is its maximal unramified abelian extension, generated over $F$ by the $j$-invariants of CM elliptic curves with endomorphism ring the full ring of integers of $F$. Merel's uniform boundedness theorem (1996) states that $\#E(K)_{\mathrm{tors}}$ is bounded by a constant depending only on the degree $[K:\mathbb{Q}]$; it is unconditional but controls torsion points rather than isogeny degrees. Momose (1995) introduced the method, based on the Galois action on differentials of covers of the $j$-line ramified only over CM points (his CM-regular covers), that Larson and Vaintrob later combined with Merel's theorem to bound prime isogeny degrees over $K$.

The expose ends with four Questions (translated from the French): (1) Is Theoreme 2 true with $C=8$? (2) Does Theoreme 5 remain valid if the hypotheses '$E$ is semistable' and '$N\geq 11$' are replaced by '$E$ has no complex multiplication' and '$N\geq 41$'? (3) If $M$ is not prime, is it true that $Y_0(M)$ has no rational point outside the known cases, printed as $M=1,4,6,8,9,10,12,,14,15,16,18,21,27$ (the printed list contains a doubled comma and omits 25, which however belongs to the intended list since $X_0(25)$ has genus 0 and infinitely many rational points)? In view of Kubert's universal bounds on torsion and of Theoreme 1, Serre notes it would suffice to treat $M=3\cdot 13$, $5\cdot 13$, $7\cdot 13$, $13^2$ and $5^3$. (4) For every number field $K$, does there exist a finite set $S_K$ of primes such that, if $N\notin S_K$, the curve $Y_0(N)$ has no rational point over $K$ apart from those coming from curves with complex multiplication? (For imaginary quadratic $K$, Goldfeld's conjectures give partial information.) Questions (1) and (3) were resolved by Kenku in 1979-1982; the surviving open core consists of Questions (2) and (4).

Problem Statement

Two of the expose's four questions were resolved by later literature: Kenku proved the optimal constant $C=8$ (Question 1) and determined the rational points on $Y_0(M)$ for composite $M$ (Question 3); see Background for the evidence. The surviving open core consists of the other two questions, which share one objective: uniform control, in the elliptic curve and in the base field, of rational points on modular curves and of the images of mod-$p$ Galois representations.

(1) Serre's uniformity problem (Question 2 of the expose). Is it true that for every elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication and every prime $p\geq 41$, the mod-$p$ Galois representation $\rho_{E,p}:G_{\mathbb{Q}}\to\mathrm{GL}_2(\mathbb{F}_p)$ is surjective? By the published reductions, for every $p\geq 41$ this is equivalent to: every rational point of the modular curve $X_{ns}^+(p)$ attached to the normalizer of a non-split Cartan subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$ is a cusp or a complex multiplication point.

(2) Uniformity over number fields (Question 4 of the expose). For every number field $K$, does there exist a finite set $S_K$ of primes such that, for every prime $N\notin S_K$, every $K$-rational point of $Y_0(N)$ is a complex multiplication point (that is, every pair $(E,A)$ over $K$ that the point classifies has $E$ with complex multiplication)? The expected affirmative answer is presently established only under the generalized Riemann hypothesis and only for fields not containing the Hilbert class field of an imaginary quadratic field; an unconditional answer for any $K\neq\mathbb{Q}$ is not known.

A counterexample to (1) is a curve $E$ over $\mathbb{Q}$ without complex multiplication together with a prime $p\geq 41$ for which $\rho_{E,p}$ is not surjective; a counterexample to (2) is a number field $K$ together with infinitely many primes $N$ admitting a non-CM $K$-rational point on $Y_0(N)$.

References

1. Jean-Pierre Serre, Points rationnels des courbes modulaires $X_0(N)$ [d'apres Barry Mazur], Seminaire Bourbaki 1977/78, expose no 511, Lecture Notes in Mathematics 710, Springer, pp. 89-100, DOI 10.1007/bfb0069974, https://www.numdam.org/item/SB_1977-1978__20__89_0/
2. Jean-Pierre Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Inventiones Mathematicae 15 (1972) 259-331, DOI 10.1007/bf01405086, https://doi.org/10.1007/bf01405086
3. Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Publications Mathématiques de l'IHÉS 54 (1981) 123-201, DOI 10.1007/bf02698692, https://doi.org/10.1007/bf02698692
4. B. Mazur, Modular curves and the Eisenstein ideal, Publications Mathématiques de l'IHÉS 47 (1977) 33-186, DOI 10.1007/bf02684339, https://doi.org/10.1007/bf02684339
5. B. Mazur and D. Goldfeld, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Inventiones Mathematicae 44 (1978) 129-162, DOI 10.1007/bf01390348, https://doi.org/10.1007/bf01390348
6. Daniel Sion Kubert, Universal bounds on the torsion of elliptic curves, Proceedings of the London Mathematical Society (3) 33 (1976) 193-237, DOI 10.1112/plms/s3-33.2.193, https://doi.org/10.1112/plms/s3-33.2.193
7. D. Goldfeld, Conjectures on elliptic curves over quadratic fields, in: Number Theory Carbondale 1979, Lecture Notes in Mathematics 751, Springer, pp. 108-118, DOI 10.1007/bfb0062705, https://doi.org/10.1007/bfb0062705
8. M. A. Kenku, On the number of Q-isomorphism classes of elliptic curves in each Q-isogeny class, Journal of Number Theory 15 (1982) 199-202, DOI 10.1016/0022-314X(82)90025-7, https://doi.org/10.1016/0022-314X(82)90025-7
9. M. A. Kenku, The modular curve $X_0(39)$ and rational isogeny, Mathematical Proceedings of the Cambridge Philosophical Society 85 (1979) 21-23, DOI 10.1017/s0305004100055444, https://doi.org/10.1017/s0305004100055444
10. M. A. Kenku, The modular curve $X_0(169)$ and rational isogeny, Journal of the London Mathematical Society (2) 22 (1980) 239-244, DOI 10.1112/jlms/s2-22.2.239, https://doi.org/10.1112/jlms/s2-22.2.239
11. M. A. Kenku, The modular curves $X_0(65)$ and $X_0(91)$ and rational isogeny, Mathematical Proceedings of the Cambridge Philosophical Society 87 (1980) 15-20, DOI 10.1017/s0305004100056462, https://doi.org/10.1017/s0305004100056462
12. M. A. Kenku, On the modular curves $X_0(125)$, $X_1(25)$, $X_1(49)$, Journal of the London Mathematical Society (2) 23 (1981) 415-427, DOI 10.1112/jlms/s2-23.3.415, https://doi.org/10.1112/jlms/s2-23.3.415
13. Loïc Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Inventiones Mathematicae 124 (1996) 437-449, DOI 10.1007/s002220050059, https://doi.org/10.1007/s002220050059
14. Fumiyuki Momose, Isogenies of prime degree over number fields, Compositio Mathematica 97 (1995) 329-348, Numdam identifier CM_1995__97_3_329_0, https://www.numdam.org/item/CM_1995__97_3_329_0/
15. Yuri Bilu and Pierre Parent, Runge's method and modular curves, International Mathematics Research Notices (2011), no. 9, 1997-2027, DOI 10.1093/imrn/rnq141, arXiv:0907.3306, https://arxiv.org/abs/0907.3306
16. Yuri Bilu and Pierre Parent, Serre's uniformity problem in the split Cartan case, Annals of Mathematics (2) 173 (2011) 569-584, DOI 10.4007/annals.2011.173.1.13, https://doi.org/10.4007/annals.2011.173.1.13
17. Yuri Bilu, Pierre Parent, and Marusia Rebolledo, Rational points on $X_0^+(p^r)$, Annales de l'Institut Fourier 63 (2013), no. 3, 957-984, DOI 10.5802/aif.2781, arXiv:1104.4641, https://arxiv.org/abs/1104.4641
18. Aurélien Bajolet, Yuri Bilu, and Benjamin Matschke, Computing integral points on $X_{ns}^+(p)$, Algebra & Number Theory 15 (2021), no. 3, 569-608, DOI 10.2140/ant.2021.15.569, arXiv:1212.0665, https://arxiv.org/abs/1212.0665
19. David Zywina, On the possible images of the mod ell representations associated to elliptic curves over Q, arXiv:1508.07660, https://arxiv.org/abs/1508.07660
20. Samuel Le Fourn and Pedro Lemos, Residual Galois representations of elliptic curves with image contained in the normaliser of a nonsplit Cartan, Algebra & Number Theory 15 (2021), no. 3, 747-771, DOI 10.2140/ant.2021.15.747, arXiv:2002.02714, https://arxiv.org/abs/2002.02714
21. Lorenzo Furio and Davide Lombardo, Serre's uniformity question and proper subgroups of $C_{ns}^+(p)$, arXiv:2305.17780, final version to appear in Algebra & Number Theory, https://arxiv.org/abs/2305.17780
22. Eric Larson and Dmitry Vaintrob, Determinants of subquotients of Galois representations associated with abelian varieties, Journal of the Institute of Mathematics of Jussieu 13 (2014) 517-559, arXiv:1110.0255, DOI 10.1017/s1474748013000182, https://arxiv.org/abs/1110.0255
23. Eric Larson and Dmitry Vaintrob, On the surjectivity of Galois representations associated to elliptic curves over number fields, Bulletin of the London Mathematical Society 46 (2014) 197-209, arXiv:1204.0046, DOI 10.1112/blms/bdt081, https://arxiv.org/abs/1204.0046
24. Barinder S. Banwait, Explicit isogenies of prime degree over quadratic fields, International Mathematics Research Notices 2023 (2023), no. 14, 11829-11876, DOI 10.1093/imrn/rnac134, https://doi.org/10.1093/imrn/rnac134
25. Barinder S. Banwait, Filip Najman, and Oana Padurariu, Cyclic isogenies of elliptic curves over fixed quadratic fields, Mathematics of Computation 93 (2024), no. 346, 841-862, DOI 10.1090/mcom/3894, https://doi.org/10.1090/mcom/3894
26. Barinder S. Banwait and Maarten Derickx, Explicit isogenies of prime degree over number fields, Algebra & Number Theory 19 (2025), no. 6, 1147-1197, DOI 10.2140/ant.2025.19.1147, https://doi.org/10.2140/ant.2025.19.1147
27. Ivan Novak, Degrees of isogenies over prime degree number fields of non-CM elliptic curves with rational j-invariant, Journal of Number Theory 278 (2026) 694-714, DOI 10.1016/j.jnt.2025.05.007, https://doi.org/10.1016/j.jnt.2025.05.007
28. Garen Chiloyan and Álvaro Lozano-Robledo, A classification of isogeny-torsion graphs of Q-isogeny classes of elliptic curves, Transactions of the London Mathematical Society 8 (2021), no. 1, 1-34, DOI 10.1112/tlm3.12024, arXiv:2001.05616, https://arxiv.org/abs/2001.05616
29. Alexander J. Barrios, Enrique González-Jiménez, and Ivan Novak, Isogeny graphs of elliptic curves in characteristic zero, arXiv:2608.02494, https://arxiv.org/abs/2608.02494
30. Maarten Derickx, Sachi Hashimoto, Filip Najman, and Ari Shnidman, Rational points on modular curves: parameterization and geometric explanations, arXiv:2602.20964, https://arxiv.org/abs/2602.20964
31. Lorenzo Furio, Effective bounds for adelic Galois representations attached to elliptic curves over the rationals, arXiv:2412.10340, https://arxiv.org/abs/2412.10340
32. Sachi Hashimoto, Guido Maria Lido, Davide Lombardo, Nicolas Mascot, and Pierre Parent, Equationless quadratic Chabauty for non-split Cartan modular curves, arXiv:2608.10919, https://arxiv.org/abs/2608.10919